Decomposition-Coordinating Method for Parallel Solution of a Multi Area Combined Economic Emission Dispatch Problem

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ISSN: 2088-8708, DOI: 10.11591/ijece.v6i5.10259  2048

Decomposition-Coordinating Method for Parallel Solution of a Multi Area Combined Economic Emission Dispatch Problem

Senthil Krishnamurthy, Raynitchka Tzoneva

Center for Substation Automation and Energy Management Systems, Department of Electrical, Electronics and Computer Engineering, Cape Peninsula University of Technology, South Africa

Article Info ABSTRACT

Article history:

Received Feb 4, 2016 Revised Jun 30, 2016 Accepted Jul 20, 2016

Multi-area Combined Economic Emission Dispatch (MACEED) problem is an optimization task in power system operation for allocating the amount of generation to the committed units within the system areas. Its objective is to minimize the fuel cost and the quantity of emissions subject to the power balance, generator limits, transmission line and tie-line constraints. The solutions of the MACEED problem in the conditions of deregulation are difficult, due to the model size, nonlinearities, and the big number of interconnections, and require intensive computations in real-time. High- Performance Computing (HPC) gives possibilities for the reduction of the problem complexity and the time for calculation by the use of parallel processing techniques for running advanced application programs efficiently, reliably and quickly. These applications are considered as very new in the power system control centers because there are not available optimization methods and software based on them that can solve the MACEED problem in parallel, paying attention to the existence of the power system areas and the tie-lines between them. A decomposition-coordinating method based on Lagrange’s function is developed in this paper. Investigations of the performance of the method are done using IEEE benchmark power system models.

Keyword:

Decomposition-coordinating method

Interconnected power systems Lagrange’s method

Multi-area combined economic emission dispatch problem Parallel computing Power system deregulation

Corresponding Author:

Senthil Krishnamurthy,

Center for Substation Automation and Energy Management Systems, Departement of Electrical, Electronics and Computer Engineering, Cape Peninsula University of Technology,

P.O.Box: 1906, Bellville, South Africa -7535.

[email protected]

1. INTRODUCTION

Deregulation of the electricity industry has been deployed in many countries to improve the economic efficiency of power system operation [1]. Electric utility systems are interconnected to achieve high operating efficiency and to produce cheap electricity with minimum production cost, maximum reliability, and better operating conditions [2]. The term multi-area power system stands for the interconnected power system. In a multi-area power system, generation and loads are coordinated with each other through the tie-lines among the areas. A load change in any one of the areas is taken care of by all generators in all areas. Similarly, if a generator is lost in one control area, governing action from generators in all connected areas will increase generation outputs to make-up the mismatch [3].

The objective of Multi-Area Combined Economic Emission Dispatch (MACEED) problem is to determine the amount of the power generated by each generator in a system and the power transfer between the areas so as to minimize the total generating cost without violating the limitations on the power produced by the generators and on the amount of tie line power transfer. In a multi-area power system, the actual local generation may not be balanced with the local demand due to the import and export of power in various

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areas. Areas of individual power systems are interconnected to operate with maximum reliability, reserve sharing, improved stability and less production cost than when they operate as isolated areas. Consideration of the transmission capacity among the areas in the multi-area power system is one of the important problems in the operation of the power system, while solving the MACEED problem. Tie line power transfer limits and power demand between area’s are considered as additional constraints in the MACEED problem. Hence the MACEED is considered as a large scale optimisation problem. MACEED is a complex problem with many different formulations and it has been considered following the development of the competitive electricity markets and the smart grid technologies. New formulation of MACEED problem is proposed in this paper.

The recent methods to solve the multi-area power system problems are based on decomposition of the overall power system problem into subproblems corresponding to the power system areas and a coordinator which are solved in an iterative way. Currently available decomposition techniques assume that the models and control objectives of the areas are formulated to be non-overlapping as it was mentioned in [4], i.e., the border of one area is at the same time also the border of a neighboring area. This type of multi- area system model is considered in the paper.

Multiarea Economic Dispatch (MAED) problem is reviewed in this section according to the methods used for its solution. They are broadly classified into classical and heuristics methods.

The papers [5]-[9] used classical methods to solve the MAED problem. The paper [5], [6] and [10]

used Lagrange’s algorithm for MAED problem solution. The papers [5] and [6], proposed an approach to incorporate power contracts into multi-area unit commitment and solve the economic dispatch solutions using an adaptive Lagrangian method. Combined Economic Emission Dispatch (CEED) problem for interconnected grids is investigated in [8]. Emissions and costs of generation are objective functions, and the interconnected grid is divided into several sub-grids. The calculation of each subarea problem is performed in parallel. A strategy is proposed that each sub-area sets different multi-objective function according to different conditions/constraints using a Linear Weighted Sum Method (LWSM). LWSM is used to change the multi-objective problem into single objective problem and it gives a preference degree of decision makers to the objective functions. The direct search method is extended to facilitate economic sharing of generation and reserve across areas and minimize the total generation cost in the multi-area reserve constrained economic dispatch in [11]. Jacobian based algorithm is used to calculate penalty factors for an area in a multi-area power system in [7]. The paper [9] developed a multi-area generation scheduling scheme that can provide proper unit commitment in each area, and effectively preserve the tie-line constraints using an improved dynamic programming method.

The research papers [9], [12]–[19] used heuristic methods to solve the MAED problem. The paper [12], presents a decomposition approach to multiarea generation scheduling problem using expert system. It proposes a large-scale mixed integer-nonlinear optimization process to solve the problem using a two-layer decomposition technique. In the first layer of decomposition, the problem is divided into several sub- problems during the considered period. The information that the problem sends to each sub-problem is the load demands of all areas at the corresponding time period and the output of the sub-problem is the system operation cost at that time. The coordination factor of this layer of decomposition is the operation cost of the system in the given period, which should be minimum. The second layer of decomposition divides the previous sub-problems further according to the control areas in the power pool. The sub-problem for each area receives system Lagrange multiplier λ and returns the area multiplier λ. The coordination at this level uses the difference between the system multiplier λ and the area multiplier λ which should be zero except for areas that reach their generation limits.

A Particle Swarm Optimisation (PSO) algorithm to solve various types of economic dispatch (ED) problems in power systems such as, multi-area ED with tie line limits, ED with multiple fuel options, combined environmental economic dispatch, and the ED of generators with prohibited operating zones using both the PSO method and the Classical Evolutionary Programming (CEP) approach is described in [13]. In the PSO method, there is only one population in each iteration that moves towards the global optimal point.

This is unlike the CEP method, which has to deal with two populations, the parents and the children, in each iteration. This makes the PSO method computationally faster in comparison to the CEP method.

In [14] and [15], MACEED problem is investigated to address the environmental issue of the economic dispatch problem. The MACEED problem is first formulated and then an Improved Multi- objective Particle Swarm Optimization (MOPSO) algorithm is developed to derive a set of Pareto-optimal solutions. Each Pareto- optimal solution is a tradeoff between operational cost and pollutant emissions. The paper [16] reviews and compares some evolutionary techniques for Multi Area Economic Dispatch (MAED) problem solutions using i) Classical Differential Evolution (DE), ii) Classical Particle Swarm Optimization (PSO), and iii) An improved PSO with a parameter automation strategy having Time Varying Acceleration Coefficients (PSO_TVAC).

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Fuzzified Particle Swarm Optimization (FPSO) algorithm for solving the security-constrained multi- area economic dispatch problem for an interconnected power system is investigated in [17] and [18]. The FPSO algorithm is based on the combined application of fuzzy logic strategy incorporated in the particle swarm optimization algorithm.

MACEED and SACEED problem is solved using PSO method in [19] and [20] respectively.

SACEED problem with valve point is considered in [21]. The gravational search method is used in [22] and [23] to solve the economic dispatch problem with multiple generator systems with considering the prohibited operating zones.

In deregulated power system environment [24] and [25] it is essential to formulate and solve the economic dispatch problem for multi-area cases with tie line constraints. The new structure of the power system requires new optimisation methods based on the specifics of this structure and providing fast, reliable and relevant to the requirements of the power system as a whole and to its elements solutions. These solutions can give deeper insight into the dynamics of the system. Methods using various types of decomposition and coordination approaches in solution of the power dispatch problems are capable to respond to the above requirements.

The existing situation is that most of the conventional gradient and heuristic methods are time consuming and still use a sequential method to solve the MACEED problem [14]-[17], [26], and [27]. The traditional heuristic methods for MACEED problem do not always guarantee global best solutions; they often achieve a fast and near global optimal solution [1],[2],[14]-[19]. Researches have constantly observed that all these methods very quickly find a good local solution but get stuck there for a number of iterations without further improvement, sometimes causing premature convergence.

Therefore, this paper develops an efficient algorithm in dealing with a large-scale MACEED problem using the decomposition approach. In comparison to lambda, direct, dynamic, Jacobian and heuristic search techniques, the Lagrange’s gradient method is more powerful tool for calculation of the complex optimisation problems. This method generally begins with an initial feasible solution and refines the solution repeatedly until the optimal solution is found.

Lagrange’s decomposition-coordinating method and algorithm are developed for multi-area economic dispatch problem solution in Parallel MATLAB environment using a Cluster of Computers. The function of Lagrange is decomposed in a number of sub-functions of Lagrange according to the number of areas by using the values of the Lagrange variables as coordinating ones. Then the initial problem is decomposed in areas sub-problems and a coordinating sub-problem. The optimal solution of the coordinating sub-problem determines the optimal solutions of the areas sub-problems and the optimal solution for the initial multi-area problem. A four area three generator and a four area four generator IEEE bench mark models are used to test and validate the results obtained by the developed software in the MATLAB Cluster of Computers.

This paper formulates the MACEED problem in section 2, Lagrange's decomposition-coordinating method and algorithm for solution of the MACEED problem are described in section 3 and 4 respectively, single area and multi-area CEED problem solutions for 4 area 10 generator system and 4 area 12 generator systems are presented in section 5. The conclusion is given in section 6.

2. MATHEMATICAL FORMULATION OF THE MACEED PROBLEM

A schematic diagram of a multiarea power system is shown in Figure 1. The system has M areas.

Every area has its own set of generatorsN_m,m1,M. The areas are interconnected by tie-lines as shown in Figure 1.

Figure 1. Model of a Multi-Area power system with tie-line power transfer

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Single criterion or multi-criteria dispatch problems can be formulated for every area. Multi-criteria combined dispatch problem is considered in the paper for every area: Two types of criteria are considered:

Type one: Operational cost

The total operational cost is the sum of the generation cost and the cost of transmission of the power between the areas, as follows: The operational cost for power production in all areas

 

^ ^{ }_ _



² ^ ^



1 1 Nm M

C mn mn mn mn mn

m n

F P a P b P c

(1) where

M is the number of the interconnected areas

Nm is the number of generators belonging to area^m^1,M and committed to the power production in this area Pmn is the real power produced by the n^th generator in the m^th area

 

  ¹ ² ³ 

. . . .

m

T

m m m m m N

P P P P P

is the vector of the real power produced in the m^th area, and m 1,M

 

  ¹ ² ³... _M^T

P P P P P is the vector of the real power produced in the whole power system amn,bmn,cmn are the cost coefficients for the power produced by the n^th generator in the m^th area.

The operational cost for transmission of the real power through the tie-lines is given as:

   

1 1

 

 ^M ^M 

T T mj Tmj jm Tjm

m j j m

F P q P q P (2)

Where

PTmjis the real power flow from the area m to the area j and PTjmis the real power flow from the area j to the area m

  

 

 ^, ¹ ^, ² ^, ³... ^, ^T, 1,

Tm Tm m Tm m Tm m Tm M

P P P P P m Mis the vector of the real power transmission between the m^th area and all other area.

 

  ¹ ² ³... ^, ¹^T

T T T T T M

P P P P P is the vector of the real power transmission between all areas.

Type two: Emission quantity

An additional criterion expressing minimisation of the pollutant emission is considered. This criterion refers only to the power generation. The tie-lines are not considered in this case because the transmission of the power does not create chemical pollution. Quantity of the pollutant emission caused by the power production is expressed as:

  

²



1 1

 

  ^M ^N^m  

E mn mn mn mn mn

m n

F P d P e P f (3)

Where:

dmn,emn,fmn are the emission coefficients of the n^th generator in the m^th area.

The combined solution of the real power dispatch problem for the multi-area system determines the optimal power to be produced by the generators in every area and the optimal values of the power transferred between the areas through the tie-lines.

The mathematical formulation of the MACEED problem criterion is based on introduction of a price penalty factors to convert the emission values of the criterion (3) into cost values and the dispatch problem to be described by a single criterion^{F P} _.

Various types of price penalty factors are proposed in [28]. Derivations in the paper are based on the min/max penalty factor, as follows:

Nm

M 2

mn mn,min mn mn,min mn

mn 2

mn mn,max mn mn,max mn m 1 n 1

a P b P c

h [$ / kg]

d P e P f

 

 





 

(5)

Where

m m1 m2 mNm

h  h h .... h is the vector of the penalty factors for the m^tharea, m1,M

1 2 M

h h h .... h is the vector of the penalty factors for the whole power system.

,min

Pmn and P_mn_,max are the minimum and the maximum real power that can be produced by the n^th generator in the m^th area.

Then the MACEED problem is formulated in the following way: Find the vector of the real power P and the vector of the power transmitted through the tie-lines PT such that the criterion (4) is minimised under the following constraints for

 

2

1 1

1

2 1

 





 

     

 

  

 

   

 

 

 



m

N M

mn mn mn mn mn mj Tmj jm Tjm

M n j

j m

m N

mn mn mn mn mn mn n

a P b P c q P q P

F P

h d P e P f

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i. Minimum and maximum real power produced by every generator

,min  ,max, 1, , 1,

mn mn mn m

P P P n N m M (5)

ii. Minimum and maximum active power sent through the tie-lines

,min  ,max, 1,

Tm Tm Tm

P P P m M (6)

These limits are valid for the two directions of the power flow and can be written as

,min ,max

1, , , 1,

    

Tmj Tmj Tmj

Tjm Tjm Tjm

P P P j M j m m M

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iii. Power balance

The balance between the power production and the power demand for the m^th area and for the whole system is given by Equation (8),

 

1 1

1 , 1,

 



 

       

^m 

N M

mn Dm Lm Tmj jm Tjm

n j

j m

P P P P  P m M (8)

Where

 

  

   0  00 

1 1 1

, 1,

m m m

N N N

Lm mn mnr mr m n mn m

n r n

P P B P B P B m M

(9)

Where

0 00

, ,

mnr m n m

B B B are the transmission loss coefficients of the interconnected power system



_jmis the fractional loss rate from the area j to the area m

PLmis the real power loss of the n^th transmission line in the m^th area.

PDmis the real power demand in the m^th area.

PTmjis the real power flow from the area m to the area j PTjmis the real power flow from the area j to the area m Then equation (8) can be written as follows:

 



^



    



 

         

    0 00 

1 1 1 1 1

1 , 1,

m m m m

N N N N M

mn Dm mn mnr mr m n mn m Tmj jm Tjm

n n r n j

j m

P P P B P B P B P P m M (10)

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The formulated problem given by Equations (1 to 10) is characterised as:

 A combined optimisation dispatch problem for every area.

 The price penalty factors are introduced for every generator separately.

 The areas are interconnected by tie-lines with two direction of real power transfer.

 The optimisation dispatch problem for the interconnected power system is multicriterial one.

 Dimension of the interconnected problem depends on the number of areas, number of the generators and the tie-lines.

Calculation of the solution of the multicriterial interconnected problem is difficult and time consuming. If the solution has to be done often and in real-time, then better computational approaches are needed. One of them is a parallel computing of the area's sub-problems and coordination of the obtained areas solutions using a High Performance Parallel Computing (HPPC) environment (Cluster of computers).

Figure 2 shows the structure of the Cluster of Computers working in MATLAB parallel computing software environment used to implement the optimisation algorithms. Parallelization of the solution is done through decomposition of the MAEED problem according to the power system interconnected areas and coordination of the obtained solutions for every area by a coordinator.

This approach requires:

i. A decomposition-coordinating method to be developed in order to obtain an algorithm for parallel calculation.

ii. Software development based on the above algorithm allowing allocation of the separate sub-problems to the structure of the HPPC environment – A Cluster of Computers.

iii. Implementation of the developed software in the HPPC environment, investigations of the capabilities of the developed algorithm.

iv. Evaluation and verification of the obtained solution by comparison of the results obtained by sequential and parallel ones.

Figure 2. Structure of the MATLAB Parallel Computing Toolbox

The proposed method is based on classical Lagrange’s optimisation [28]-[31]. The literature review found that the classical Lagrange's method has been used till now only for sequential solution of the MACEED problem in [5],[8],[13]-[18], and [20]. The Lagranges method is introduced in early 1972 for the power system spinning reserve determination in a multi system configuration [32], an interim multi-area economic dispatch [33], and two area power systems economic dispatch problem in [34].

The paper introduces a parallel solution of the MAEED problem by considering two-level calculation structure, Figure 3, where the initial optimisation problem is decomposed into sub-problems (for every area one), solved on the first level, and the obtained solutions are coordinated by a coordinating sub- problem on the second level - in order to obtain the global solution of the whole problem.

Implementation of the MAEED problem in real-time is done in the following way, Figure 3: All data from the separate areas are sent using the communication system to the main control center where the Cluster of computers is located. The problem is solved and the optimal solutions are sent to the areas to be used for control of the generators power production. This scenario can be repeated through some selected periods of time, for example every hour.

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1 m

 

P opt1

1( )

PD dem P optm 

PDm PDM PMopt

2 M

P1

P2 Pm P_M

where : λ1 – coordinating variable; (opt) - optimal; (dem) - demand Figure 3. Real-time implementation structure of the MAEED problem solution

3. DEVELOPMENT OF LAGRANGE’S DECOMPOSITION-COORDINATING METHOD FOR SOLUTION OF THE MACEED PROBLEM

Development of the method is based on construction of a function of Lagrange for the multi-area dispatch problem, as follows:

     

 

2 2

1 1 1 1 1 1

0 00

1 1 1 1 1

1

     



    



       

   

   

           

 

   

   

m m

m m m m

N N

M M M M

mn mn mn mn mn mn mn mn mn mn mn mj Tmj jm Tjm

m n m n m j

j m

N N N N M

m mn Dm mn mnr mr m n mn m Tmj jm Tjm

n n r n j

j m

a P b P C h d P e P f q P q P

L

P P P B P B P B P P

 

1

 

 

 



^M

m

(11)

Where:



1 2 .... _m.... _M



      is the vector of the Lagrange’s multipliers.

It is necessary to find the values of _P_mn_,_P_Tmj_,_P_Tjm_and__m_,_m__1,_M such that the function of Lagrange has an optimum solution which is minimum according to P_mn,P_Tmj_,P_Tjm , and maximum according to _m, under the constraints.

,min ,max, 1, , 1,

mn mn mn m

P P P m M n N

(12)

,min   ,max, 1, , 1, , 

Tmj Tmj Tmj

P P P m M j M j m

(13)

,min   ,max, 1, , 1, , 

Tjm Tjm Tjm

P P P m M j M j m

(14) The optimal solution is found on the basis of the necessary conditions for optimality according to

the real powers and according to the Lagrange’s multipliers as follows:





 

         

 



⁰ 

1

2 2 1 2 0

Nm

mn mn mn mn mn mn mn mn m mnr mr m n

mn r

L a P b h d P h e B P B

P

(15) Where ⁿ^^1,^N^m^,^{and m}^^1,^M

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0, 1, , 1, ,

       

 _Tmj ^mj ^m ^pTmj

L q e m M j M j m

P 

(16)



¹



^0, ^1, ^, ^1, ^,

        

_Tjm ^jm ^m ^jm ^pTjm

L q e m M j M j m

P  

(17)

Where e_pTmj,and e_pTjmare the values of the corresponding derivatives.









 _ _ _ _ _



            



  

⁰ ⁰⁰



1 1 1 1 1

1 0

m m m m

N N N N M

mn Dm mn mnr mr m n mn m Tmj jm Tjm m

m n n r n j

j m

L P P P B P B P B P P e

(18) wheree__mis the value of the derivative, m1,M j, 1,M j, m

Solution of the system of Equations (15) – (18) can be achieved in a hierarchical calculation structure using decomposition approach based on selection of coordinating variables [28]. These are selected to be _m,m1,M

It is supposed that the values of the coordinating variables are given by the coordinator in the two level hierarchical structure of calculations, Figure 3 as follows:

, 1,

l

m mm M

   (19)

Where l is the index of the iterations on the coordinating level

When _m is known the Equation (15) can be solved in a decentralized way on a first level of the calculation structure and the Equations (16) to (18) can be solved on the second level for the obtained on the first level variables P_mn

Derivation of the solution of equation (15) is as follows: Equation (15) is transformed to express explicitly all terms of P_mn:



 

           

  



⁰

1

2 2 2 2 1 0

Nm

mn mn m mnn mn mn mn mn mn mn mn m mnr mr m n

mn r

r n

L a P B P b h d P h e B P B

P  

(20) If _m 0and is given by the coordinator Equation (20) can be written as:



   

           

  



 ⁰ 

1

1 1 0

2

Nm

mn mn mn mn mn mn

mnn mn mnr mr m n

mn m m m m m

m n

a h d b h e

L B P B P B

P    

(21) From here



      

    

   

 



 ⁰  

1

1 1,

1 ,

2 1,

Nm

mn mn mn mn mn mn

mnn mn mnr mr m n

m rr n m m

m M

a h d b h e

B P B P B

n N

 

(22) Equation (22) can be written in a vector matrix form in Equation (23)

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 

    

 

   

     

      

    

   

     

    

 

 

1 1 1 1 1 1

11 12 1 01

1

2 2 2 2 2 2

21 22 2 2 02

1 2

.. 1

... 1

1 : 2

: : : : :

.. 1

m

m m

m m mm

m m m m m m

m m m N m

m m

m

m m m m m m

m m m N m m

m m

mN

mN m m mN m m

mN mN nN N

m m

a h d b h e

B B B B

P

a h d b h e

B B B P B

a h d P b h e

B B B

 

 

 

  

 

 ⁰ 

, 1,

n Nm

m M

B

(23) In a short form Equation (23) can be written as

, 1,

m m m

E P D m M (24)

If the value of _m^l,m1,M is known, then the solution for the active power of the m^th area for the l^th iteration is

\ , 1,

 

l l l

m m m

P E D m M (25)

Solution of the equations (16) and (17) can be done by gradient procedures as follows:

a) Initial values of P_Tmj^q^m and P_Tjm^q^m are guessed

, 1, 1

 ^s^m  ^g^m  

Tmj Tmj Tjm Tjm m m

P P and P P s g

Where s_m 1,k and _m ^g^m^^1,^k^m are the indexes of the gradient procedures in the m^th area for the tie-line powers P_Tmj^s^m and P_Tjm^g^m respectively,k_m_,m1,M are the maximum number of iteration for the m^tharea.

The value of k_mcan be different for each of the areas.

b) The improved values of the tie-lines real power are

1 

m m m

s s s

Tmj Tmj Tmj PTmj

P P  e

(26)



¹  ^m

m m

PTjm

g

g g

Tjm Tjm Tjm

P P e

(27)

Where _Tmj and _Tjm are the steps of the gradient procedures.

sm

ePTmj and e_PTjm^g^m are given by Equations (16) and (17), m1,M j, 1,m j, m The calculation of the values of the _P_Tmj^s^m^¹_{and P}_Tjm^g^m^¹stops when

 1

m PTmj

es  and ^m _ ₂

PTjm

eg  (28)

Where: ₁_and₂ are very small positive numbers.

Solutions (25), (26) and (27) depend on _m,m1,M.

When the optimal value of _mis obtained, the values of P_mn_,P_Tmjand P_Tjmwill reach their optimum. A gradient procedure is used to calculate the optimal value of _m,m1,M, as follows:

1 , 1,

   

l l l

m m _me_m m M

   (29)

Where __m,m1,M are the steps of the gradient procedure and e^l__m_,m_1,Mis given by equation (18). The gradient procedure on the second level stops when

(10)

3, 3 0, 1,

  

l

e_m   m M (30)

Or when l reaches the maximum number of iterations ^L__m.

4. ALGORITHM OF THE LAGRANGE’S DECOMPOSITION-COORDINATING METHOD FOR CALCULATION OF THE MAEED PROBLEM

The steps of the Lagrange’s decomposition-coordinating method are:

1. Values of all coefficients are given and the number of iterations of the second L__m



^{k m}^m^, ^^1,^M



^are

given.

2. Initial values of the coordinating variables _m^l,m1,M are guessed, l=1 3. Initial values of the tie-lines active powers are guessed

, , 1, , 1, , 

m m

Tmj Tjm

s g

P P m M j M j m

4. Calculation on the first level are done for every subsystem m1,M using Equation (25).

5. The solutions of the first level P_mn^l ,n1,N_m,m1,Mare checked according to the constraints (12), and are sent to the second level.

6. On the second level the values of the tie-lines are calculated by gradient procedures:

i. Equations (26) and (13) are solved ii. Equations (27) and (14) are solved

The gradient procedures (26) and (27) stop when the conditions (28) are fulfilled respectively or the maximum number of iterations k_m,m1,M is reached.

7. The solutions of the second level for ^l

PTmj and ^l

PTjm,m1, ,M j1,M ,jm are checked according to the constraints (13) and (14).

8. On the second level the improved values of the Lagrange’s variables are determined. Equations (18) and (29) are calculated. The gradient procedure (29) stops when the condition (30) is fulfilled or the maximum number of iterations L__m is reached. In this case the optimal solution for _m,m1,M is obtained and the corresponding to it optimal solutions of the problems on the first and second levels are obtained.

The iterations on the second level for calculation of __m_,_m_1,_Mcan stop before reaching the optimum solution if a maximum number of iteration L__mis determined.

The hierarchical calculating structure is shown in Figure 4. In the above algorithm for one-step of the gradient procedure for optimisation of _mon the 2^nd level, k_mmaximum number of iterations on the second level are performed to obtain the optimal solutions for the tie-line real powers and one calculation of the generators real power is done on the first level.

1

1 ,n n1,N1

P ^m

M , 1,

mn m

P n N P_Mn,n1,N_M

Figure 4. Hierarchical structure for the MAEED problem solution using the developed Lagrange’s decomposition-coordinating algorithm

5. STUDIES OF THE SINGLE AREA AND MULTI-AREA CEED PROBLEM SOLUTIONS The proposed algorithm is tested using two benchmark models of single and multi-area power systems. They are:

(i) Four areas with ten generators in each area without considering the transmission line losses (ii) Four area with three generators in each area with considering the transmission line losses

The two bench mark models are applied for two considered scenarios, they are:

(11)

a) The whole power system is considered as a single area one prior to decompose the power system into multi-areas. The tie-line constraints are not considered.

b) The whole power system is decomposed into multi-areas with tie-line constraints.

5.1. Single Area Combined Economic Emission Dispatch (SACEED) problem solutions

5.1.1. Case study 1: Four areas power system with ten generators in each area without considering the transmission line losses

The fuel cost and emission data of the system are given in [32]. The generators in each area have different fuel and emission characteristics and the tie-lines have different transfer limits. The system has a total power demand of 10500 [MW]. The transmission cost is neglected in the process of problem solution since it is normally small as compared with the total fuel cost. The initial value of _mfor m=1 is assumed as 4 and the maximum number of iterations is set to 10000. The single area economic dispatch problem is solved for power demand of 10500 [MW] in a sequential way and the results are given in Table 1.

The flowchart of the Lagrange’s decomposition-coordinating algorithm for calculation of parallel solution of a MACEED problem is given in Figure 5.

00 0 ,min

,max

, , , , , , , , , , , , , , , ,

, , , , 1, , 1,

m m

l

Dm m mn mn mn mn mn mn mj

s g

mj jm mn m m n jm mn

l

mn m m

P a b c d e f

PT PT B B B P

P k L k n N m M

 



l, l m m

E D

l l l

m m m

P E \ D

m mj l,s

Calculate ePT u sin g(16)

,

 1 m mj l q

ePT 

1 

l l l

m m _me_m

  

Set sm1,m1,M

m m

s k

 _m l L_

 3 l

e_m 

1,

m M

l,

Pm P_m^l,m1,M

, 1,

l

e_m m M

m mj l, s

Calculate PT

u sin g(26)

m mj l, s T

Check the constra int s P u sin g(13)

l

m m

l   1, ,ml,M

Set gm1,m1,M

m jm l,g

Calculate ePT u sin g(17)

,

 2 m jm l q

ePT 

m jm l, q T

Check the constra int s P u sin g(14)

m jm l,g

Calculate PT

u sin g(27)

m m

g k

Figure 5. Decomposition-coordinating method flow diagram for parallel solution of the MACEED problem